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2017-03-27 Intermediate

         

Two identical jars \(A\) and \(B\) each contain a fruit fly of equal mass. The lids of the jars are tightly closed. In jar \(A\), the fly is sitting; in jar \(B\), the fly is flying at a constant speed in a horizontal circle.

The jars are placed on weighing scales to see which jar weighs more. Which jar will show a greater reading on the scale?

Amanda, Billy, Caleb, David, and Ellie are told that they are each given a distinct integer from 1 to 5 inclusive. They each know their own integer, but are not told the integer of anyone else. They make the following statements:

Amanda: "My number has an odd number of positive factors."
Billy: "Really? My number is either odd or prime, but not both."
Caleb: "I now know Amanda's number."

Given that David's number is less than Amanda's number, what number does Ellie have?

Construct a square on each side of a triangle with side lengths 3, 4, and 5. Then, construct a hexagon by joining the vertices of the squares which are not vertices of the 3-4-5 triangle.

Find the area of this hexagon.

You are given 15 letters above with which you can write my name KEVIN.
Suppose that, each time you go down one row, you have 3 choices: go straight down, or go diagonally one to the left, or go diagonally one to the right.

Then, how many ways are there to spell out KEVIN?


Bonus: Can you generalize to \(n\) rows?

Let ABCD be a convex quadrilateral. Also let P, Q, R, S be the respective midpoints of sides AB, BC, CD, DA such that triangles AQR and CSP are both equilateral.

Find the largest angle (in degrees) of quadrilateral ABCD.

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